Tarone-Ware Test

Tarone-Ware Test for Comparing Survival Curves

The Tarone-Ware test asks whether two or more groups, such as the arms of a randomised trial, share the same time-to-event experience once censoring is allowed for. It is one of the weighted log-rank tests, and its weight shrinks gently as fewer patients remain under observation, so deaths or progressions early in follow-up count for somewhat more than later ones. Health economists meet it in trial reports and appraisal submissions when survival curves separate early and a check less dependent on proportional hazards than the ordinary log-rank test is wanted. Its output is a chi-square statistic and a p-value, never a survival gain, so a cost-effectiveness model still needs estimated and extrapolated survival curves whatever the test shows.

Origin of the square-root weight

Tarone and Ware, in Biometrika in 1977, showed that the log-rank statistic and the modified Wilcoxon statistic for right-censored data differ only in their choice of weights, and that those weights are functions of the number of individuals at risk at each death. They proposed a related statistic that may be more powerful across a range of alternatives than either. The Stata manual describes the Tarone-Ware test as the one suggested in that paper, with weights equal to the square root of the number of subjects in the risk pool.

The family is easiest to see side by side. The Stata manual tabulates each test's weight at a distinct failure time, where $n_i$ is the number at risk and $\tilde S(t_i)$ is a modified estimate of the pooled survivor function.

The square root sits between a constant and the full risk-set size, which places the test between the log-rank and the Wilcoxon test in how strongly it favours early events. The Peto test and the Fleming-Harrington test build their weights from estimates of pooled survival based on the Kaplan-Meier estimator instead, so they follow the shape of the survival curve, not the size of the risk set.

The two-arm statistic

At each distinct event time the test compares the deaths observed in one arm with the number expected if both arms had the same hazard, given who is still at risk. The Stata manual gives the general form for $r$ groups, which for two arms reduces to the following.

$$U_{TW} = \sum_{j=1}^{k} \sqrt{n_j}\,\left(d_{1j} - e_{1j}\right), \qquad e_{1j} = \frac{n_{1j}\,d_j}{n_j}$$

where $k$ is the number of distinct event times, $n_j$ is the number at risk in both arms just before event time $t_j$, $n_{1j}$ is the number at risk in arm 1, $d_j$ is the total number of events at $t_j$, $d_{1j}$ is the number of those events in arm 1, and $e_{1j}$ is the number of arm 1 events expected under the null hypothesis of equal hazards.

$$V_{TW} = \sum_{j=1}^{k} n_j\,\frac{n_{1j}\,n_{0j}\,d_j\,(n_j - d_j)}{n_j^{2}\,(n_j - 1)}, \qquad \chi^2_{TW} = \frac{U_{TW}^{2}}{V_{TW}}$$

where $n_{0j}$ is the number at risk in arm 0, the factor $n_j$ in front is the squared weight $(\sqrt{n_j})^2$, $V_{TW}$ is the variance of $U_{TW}$ under the null hypothesis, and $\chi^2_{TW}$ is the test statistic. For two arms it is compared with a chi-square distribution with one degree of freedom, and for $r$ groups the Stata manual gives $r - 1$ degrees of freedom. When only one event occurs at a time, each term of $V_{TW}$ simplifies to $n_{1j} n_{0j} / n_j$.

The manual also describes a stratified version, with score and variance summed over strata, and a trend test for ordered groups using the same weights. Multiplying every weight by one constant leaves $\chi^2_{TW}$ unchanged, so only relative weights matter.

What the weight does over follow-up

The weight at the last event time compared with the first is the square root of the ratio of the two risk-set sizes. In an illustrative trial whose risk set shrinks to a quarter of its starting size, the log-rank test weights first and last events equally, the Tarone-Ware test gives the first event twice the weight of the last, and the Breslow-Gehan test gives it four times the weight.

Because the weight is built from the number at risk, it moves with anything that empties the risk set. Deaths, loss to follow-up and administrative censoring of late entrants all count, so two trials with identical survival curves but different accrual can weight the same months differently. The Stata manual says the test suits hazards thought to vary other than proportionally when censoring patterns are similar across groups, and is less susceptible than the Wilcoxon test to the censoring pattern, although large differences between groups could remain a problem. The R survival documentation notes that Peto and Peto showed the Gehan-Wilcoxon test can be badly biased under different censoring patterns, and the Stata manual states that the Peto-Peto-Prentice test is not affected by them, which matters when censoring differs by arm.

When few patients have an event, the risk set barely shrinks, the square-root weights are almost constant and the result is close to the log-rank result. Royston and Parmar, simulating tests for randomised trials, found that a low control-arm event rate reduced the power of weighted log-rank tests aimed at early effects. They explain that with such data the early-effect weights vary little, so the weighted test too closely resembles the standard log-rank test, and the same arithmetic applies to square-root weights on a slowly shrinking risk set.

Choosing the test before unblinding

The Tarone-Ware test gains statistical power over the log-rank test when the true difference between arms is concentrated where its weights are largest. Royston and Parmar note that when the proportional hazards assumption is false, the log-rank test loses power and the hazard ratio becomes harder to interpret as the treatment effect, and they found that several tests aimed at a particular pattern performed well when the correct type of treatment effect had been assumed. A trial expecting early separation can prespecify the Tarone-Ware test as primary or as a sensitivity analysis. Picking the weighted test with the smallest p-value after seeing the curves inflates the false-positive rate.

Planning can use the same weights. The SAS POWER procedure computes power and sample size for two survival curves under the log-rank, Gehan or Tarone-Ware tests (TEST=TARONEWARE) by the method of Lakatos, so the answer depends on the assumed shape of both curves, accrual and loss to follow-up.

Worked example with early deaths in the control arm

All numbers in this example are illustrative. A small trial randomises 10 patients to a new treatment (arm 1) and 10 to standard care (arm 0) and follows everyone for 12 months. In arm 1 deaths occur at months 5, 9 and 10, and seven patients are alive and censored at 12 months. In arm 0 deaths occur at months 1, 2, 3 and 8, one patient is lost to follow-up at month 6, and five are censored at 12 months.

1. Build the risk sets. At each of the seven death times the table lists the numbers at risk just before the death, expected arm 1 deaths and the weighted terms. Each time has a single death, so the variance term is $n_{1j} n_{0j}/n_j^2$ and the squared-weight term is $n_{1j} n_{0j}/n_j$.

2. Sum the score. Arm 1 had 3 deaths against 4.0283 expected. The weighted score is $U_{TW} = -2.2361 - 2.2942 - 2.3570 + 1.6977 - 2.3238 + 1.3363 + 1.3868 = -4.7902$, negative because arm 1 had fewer deaths than expected.

3. Sum the variance. $V_{TW} = 5.0000 + 4.7368 + 4.4444 + 4.1176 + 3.6000 + 3.2143 + 3.0769 = 28.1901$.

4. Form the statistic. $\chi^2_{TW} = 4.7902^2 / 28.1901 = 22.9460 / 28.1901 = 0.8140$, with one degree of freedom and a p-value of 0.37.

5. Compare the neighbours. On the same data the log-rank test gives $U = -1.0283$, $V = 1.6947$ and $\chi^2 = 0.6240$ (p = 0.43), and the Breslow-Gehan test gives $U = -22$, $V = 479$ and $\chi^2 = 1.0104$ (p = 0.31). The Tarone-Ware result falls between them because three of the four control deaths came in the first three months, where its weights are largest.

6. Interpret. None of the three tests gives evidence of a difference with 20 patients, and the result says nothing about the size of any benefit. For an economic model the same data give Kaplan-Meier survival at 12 months of 0.70 in arm 1 and 0.5833 in arm 0, and a restricted mean survival time to 12 months of $\text{RMST}_1 = 5 + 4(0.9) + 0.8 + 2(0.7) = 10.80$ months against $\text{RMST}_0 = 1 + 0.9 + 0.8 + 5(0.7) + 4(0.5833) = 8.53$ months, a difference of 2.27 months within follow-up.

Survival inputs a cost-effectiveness model still needs

NICE Decision Support Unit Technical Support Document 14, on survival analysis for economic evaluations alongside clinical trials, explains why a test is not enough. Survival data are often censored, so extrapolation is needed to estimate the full survival benefit, and relying on the observed trial period alone is likely to underestimate the survival gain and the QALYs gained. The document notes that mean survival equals the area under the survival curve and that parametric models are likely to be the preferred way of bringing survival data into health economic models in most cases.

A Tarone-Ware p-value feeds none of these steps directly. It does not choose a survival distribution, supply the curves a partitioned survival model needs, or show that proportional hazards fail. TSD 14 also stresses that standard survival methods suit only uninformative censoring, a condition the Tarone-Ware test shares. In an appraisal, a significant early-weighted test alongside a non-significant log-rank test is a signal to look at time-varying hazards and at how the extrapolated curves treat the period after the early separation.

Software conventions and a common mix-up

Stata runs the test with sts test, tware, with strata() for the stratified form and trend for ordered groups. In Python, the statsmodels survdiff function takes weight_type="tw", documented as weighting by the square root of the risk-set size. In the Stata manual's heart transplant example, with little difference between groups, the chi-square values are 0.13 for the log-rank, 0.14 for Wilcoxon-Breslow-Gehan and 0.12 for Tarone-Ware.

The R survival function survdiff weights each death by $\hat S(t)^{\rho}$, where $\hat S(t)$ is the Kaplan-Meier estimate. A value of $\rho = 0.5$ weights by the square root of pooled survival, a different statistic from the Tarone-Ware test, so a report should name the weight actually used.

Limitations and common misreadings

The test is simple and widely available. Its weight carries assumptions about trial design as well as about when any effect appears.

1. Reading significance as a treatment effect. The statistic tests equality of survival distributions, and the size of any difference must come from survival estimates or a model.
2. Treating it as a test of proportional hazards. Disagreement with the log-rank result hints at where curves differ but does not diagnose the hazard pattern.
3. Ignoring the censoring pattern. Unequal loss to follow-up or accrual between arms can distort the emphasis.
4. Choosing the weight after seeing the curves. Selecting the most favourable weighted test invalidates the nominal significance level unless multiplicity is controlled, for example by hierarchical testing.
5. Expecting large gains from a mild weight. With low event rates the test behaves almost like the log-rank test, and with a delayed effect it puts its largest weights where the arms differ least.
6. Confusing implementations. The square-root weight is on the number at risk, not on pooled survival.

Sources

• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Decision Support Unit; June 2011, last updated March 2013. https://sheffield.ac.uk/sites/default/files/2022-02/TSD14-Survival-analysis.updated-March-2013.v2.pdf
• R survival package. survdiff: Test survival curve differences. Package survival version 3.8-6 help page, accessed 31 October 2026. https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survdiff.html
• Royston P, Parmar MKB. A simulation study comparing the power of nine tests of the treatment effect in randomized controlled trials with a time-to-event outcome. Trials. 2020;21:315. https://doi.org/10.1186/s13063-020-4153-2
• SAS Institute Inc. SAS/STAT 13.2 User's Guide. The POWER Procedure, Analyses in the TWOSAMPLESURVIVAL Statement. Accessed 31 October 2026. https://support.sas.com/documentation/cdl/en/statug/67523/HTML/default/statug_power_details78.htm
• StataCorp. sts test: Test equality of survivor functions. Stata Survival Analysis Reference Manual. StataCorp LLC; 2025. https://www.stata.com/manuals/stststest.pdf
• statsmodels. statsmodels.duration.survfunc.survdiff, statsmodels 0.15.0 documentation, accessed 31 October 2026. https://www.statsmodels.org/stable/generated/statsmodels.duration.survfunc.survdiff.html
• Tarone RE, Ware JH. On distribution-free tests for equality of survival distributions. Biometrika. 1977;64(1):156-160. https://doi.org/10.1093/biomet/64.1.156